How to Do Garfield's Proof of the Pythagorean Theorem

Step-by-Step Guide

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   Step 1: Construct a right triangle resting on side b with right angle to the left connected to upright and perpendicular side a

   We are interested to know the angle x formed where the the two side c's meet.
   
   Thinking about it, the original triangle was made of 180 degrees with the angle on the right at the far end of b, called theta, and the other angle at the top of a, being 90 degrees minus theta, as all the angles total 180 degrees and we already have one 90 degree angle. , At the bottom, we have  theta, at the top left we have 90 degrees, and the top right we have 90 degrees minus theta.
   
   The mystery angle x is 180 degrees.
   
   So theta + 90 degrees-theta + x = 180 degrees.
   
   Adding theta and negative theta gives us zero on the left, and subtracting 90 degrees from both sides leaves x equal to 90 degrees.
   
   So we have established that the mystery angle x = 90 degrees. , First, the formula for a trapezoid is A= the Height x (Base1 + Base 2)/2.
   
   The height is a+b and (Base1 + Base 2)/2 = 1/2(a + b).
   
   So that all equals 1/2 (a+b)^2. , We have the two smaller triangles at bottom and left, and those together equal 2*1/2(a*b), which just equals (a*b).
   
   Then we also have 1/2 c*c, or 1/2 c^2.
   
   So together we have the other formula for the area of the trapezoid equaling (a*b)+ 1/2 c^2. , 1/2(a+b)^2=(a*b)+1/2 c^2.
   
   Now multiply both sides by 2 to get rid of the 1/2's 2(1/2 (a+b)^2) = 2((a*b)+ 1/2 c^2.) which simplifies as (a+b)^2 = 2ab + c^2. , to obtain a^2 + b^2 = c^2, The Pythagorean Theorem! ,, For more art charts and graphs, you might also want to click on Category:
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   Step 2: with side c connecting the endpoints of a and b.

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   Step 3: Construct a similar triangle with side b now extending in a straight line from the original side a

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   Step 4: then with side a parallel along the top to the bottom original side b

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   Step 5: and side c connecting the endpoints of the new a and b.

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   Step 6: Understand the goal.

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   Step 7: Transfer your angle knowledge to the upper new triangle.

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   Step 8: Look at the whole figure as a trapezoid in two ways.

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   Step 9: Look at the interior of the trapezoid and add up the areas

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    Step 10: in order to set them equal to the formula just found.

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    Step 11: Set the two Area formulas equal.

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    Step 12: Now expand the left hand square

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    Step 13: which becomes a^2 + 2ab + b^2

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    Step 14: and we see we can subtract 2ab from both sides of a^2 + 2ab + b^2

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    Step 15: = 2ab + c^2.

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    Step 16: Finished!

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    Step 17: Make use of helper articles when proceeding through this tutorial: See the article Create Higher Exponential Powers Geometrically for a list of articles related to Excel

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    Step 18: Geometric and/or Trigonometric Art

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    Step 19: Charting/Diagramming and Algebraic Formulation.

Detailed Guide

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